research papers J. Appl. Cryst. (2020). 53, 927–936 https://doi.org/10.1107/S1600576720006925 927 Received 22 November 2019 Accepted 22 May 2020 Edited by S. Marchesini, Lawrence Berkeley National Laboratory, USA 1 This article will form part of a virtual special issue of the journal on ptychography software and technical developments. ‡ Present address: ARC Centre of Excellence in Advanced Molecular Imaging, School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia. § Present address: Synchrotron Radiation Research, Lund University, Box 118, 221 00, Lund, Sweden. } Present address: SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, California, USA. ‡‡ Present address: European XFEL, Holzkoppel 4, 22869 Schenefeld, Germany. §§ Present address: School of Molecular Sciences, Arizona State University, Tempe, AZ 85287, USA. }} Present address: School of Physical Science and Technology, Shanghai Tech University, Shanghai, People’s Republic of China. Keywords: X-ray speckle tracking; ptychography; wavefront metrology; X-ray optics; multi-layer Laue lenses. Ptychographic X-ray speckle tracking with multi-layer Laue lens systems 1 Andrew J. Morgan, a *‡ Kevin T. Murray, b Mauro Prasciolu, b Holger Fleckenstein, a Oleksandr Yefanov, a Pablo Villanueva-Perez, a § Valerio Mariani, a } Martin Domaracky, a Manuela Kuhn, b Steve Aplin, a ‡‡ Istvan Mohacsi, b ‡‡ Marc Messerschmidt, c §§ Karolina Stachnik, b Yang Du, a }} Anja Burkhart, b Alke Meents, b Evgeny Nazaretski, d Hanfei Yan, d Xiaojing Huang, d Yong S. Chu, d Henry N. Chapman a,e,f and Sas ˇa Bajt b,e * a CFEL, Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany, b DESY, Notkestrasse 85, 22607 Hamburg, Germany, c National Science Foundation BioXFEL Science and Technology Center, 700 Ellicott Street, Buffalo, NY 14203, USA, d National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, NY 11973, USA, e The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany, and f Department of Physics, Universita ¨t Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany. *Correspondence e-mail: [email protected], [email protected]://doi.org/10.11577/1616246https://doi.org/10.11577/ The ever-increasing brightness of synchrotron radiation sources demands improved X-ray optics to utilize their capability for imaging and probing biological cells, nano-devices and functional matter on the nanometre scale with chemical sensitivity. Hard X-rays are ideal for high-resolution imaging and spectroscopic applications owing to their short wavelength, high penetrating power and chemical sensitivity. The penetrating power that makes X-rays useful for imaging also makes focusing them technologically challenging. Recent developments in layer deposition techniques have enabled the fabrication of a series of highly focusing X-ray lenses, known as wedged multi-layer Laue lenses. Improvements to the lens design and fabrication technique demand an accurate, robust, in situ and at-wavelength characterization method. To this end, a modified form of the speckle tracking wavefront metrology method has been developed. The ptychographic X-ray speckle tracking method is capable of operating with highly divergent wavefields. A useful by-product of this method is that it also provides high-resolution and aberration-free projection images of extended specimens. Three separate experiments using this method are reported, where the ray path angles have been resolved to within 4 nrad with an imaging resolution of 45 nm (full period). This method does not require a high degree of coherence, making it suitable for laboratory-based X-ray sources. Likewise, it is robust to errors in the registered sample positions, making it suitable for X-ray free-electron laser facilities, where beam-pointing fluctuations can be problematic for wavefront metrology. 1. Introduction In 2015, Morgan et al. (2015) reported on the use of a lens for one-dimensional focusing of hard X-rays, with a photon energy of 22 keV. This lens was made by alternately depositing two materials with layer periods that follow the Fresnel zone- plate condition and then slicing the structure approximately perpendicular to the layers to the desired optical thickness. By varying the tilt of the layers throughout the stack, so that the Bragg and zone-plate conditions are simultaneously fulfilled for every layer, large focusing angles can be achieved with uniform efficiency. Such a structure is referred to as a wedged multi-layer Laue lens (MLL) (Yan et al., 2014), which is ISSN 1600-5767
Transcript
research papers
J. Appl. Cryst. (2020). 53, 927–936 https://doi.org/10.1107/S1600576720006925 927
fabricated by the use of a masked magnetron sputtering
technique, and is schematically illustrated in Fig. 1.
Errors in the wavefront produced by the wedged MLL were
characterized using a pseudo-one-dimensional ptychographic
algorithm. This analysis revealed a defect in the lens that
produced two distinct regions along the layer stack, each with
a different focal length. Further studies revealed that the
defect was caused by a transition in the material pair from
amorphous to crystalline phase for layer periods of about
5.5 nm (Bajt et al., 2018). By switching to a new material pair
(tungsten carbide and silicon carbide) the phase transition
could be avoided, allowing for a larger lens stack with greater
focusing power. This illustrates the importance of wavefront
metrology as a diagnostic tool for the iterative development of
new optical elements.
Ptychography is a powerful tool for wavefront metrology, as
it allows for the simultaneous recovery of the complex-valued
wavefront produced by the lens and the complex-valued
transmission function of the sample which is scanned across
the wavefront, typically near the focal plane of the lens, with
diffraction-limited resolution (Chapman, 1996; Rodenburg et
al., 2007; Thibault et al., 2009). The high resolution is a result
of the fact that ptychography often employs a fully coherent
model21for the wavefront propagation from the sample to the
detector plane, with few approximations beyond paraxial
illumination, a thin specimen and a high degree of coherence
of the imaging system.
However, ptychography can present difficulties in its
implementation, in part because the coherent model of the
imaging system can be sensitive to errors in the estimated
model parameters. It can also be computationally demanding
to perform the required number of iterative steps in the
reconstruction algorithm, which can be exacerbated by the
large number of diffraction patterns in some ptychographic
data sets. Furthermore, determination of the root cause of a
failed reconstruction, for example, bad detector readings,
sample stage jitter, X-ray source incoherence or algorithm
parameters, can be difficult owing to the complicated rela-
tionship between the measured diffraction intensities and the
recovered wavefronts. For example, although the wavefront
reconstruction reported by Morgan et al. (2015) took only a
few hours to complete, this calculation was preceded by many
months of work identifying detector artefacts, exploring
reconstruction parameters and checking the uniqueness of the
output.
Since the work of Berujon et al. (2012) and Morgan et al.
(2012) (no relation to the current author), X-ray speckle
tracking (XST) techniques have emerged as a viable tool for
wavefront metrology applications. This method is based on
near-field speckle-based imaging, where the 2D phase gradient
of a wavefield can be recovered by tracking the displacement
of localized ‘speckles’ between an image and a reference
image produced in the projection hologram of an object with a
random phase/absorption profile (random in the sense that the
modulation of the beam by the object is both detailed and
non-repeating over the relevant spatial frequencies of the
image). Additionally, XST can be employed to measure the
phase, absorption and ‘dark-field’ profile of an object’s
transmission function. Thanks to the simple experimental
setup, high angular sensitivity and compatibility with low-
coherence sources, this method has since been actively
developed for use in synchrotron and laboratory light sources;
see Zdora (2018) for a recent review.
As part of an ongoing project to develop and improve the
fabrication and performance of wedged MLLs for imaging
(Prasciolu et al., 2015; Murray et al., 2019), we have developed
a modified form of XST suitable for highly divergent illumi-
nation conditions (Morgan, Quiney et al., 2020). This method,
the ptychographic X-ray speckle tracking (PXST) technique,
adopts an experimental geometry and iterative update algo-
rithm similar to that employed in many ptychographic appli-
cations. Under ideal imaging conditions, the PXST method
will not achieve the same (diffraction-limited) sample-imaging
resolution or phase sensitivity that could be achieved via
ptychographic approaches. However, we show that it is
possible to recover images with large magnification factors, of
around 2000 or more, and thus PXST can provide sufficiently
high phase sensitivity and imaging resolution for many
applications. On the basis of a pseudo-geometric approxima-
tion for the propagation of light from the sample exit surface
to the detector plane, the source of errors in the recovered
wavefronts can be localized to individual intensity measure-
ments, leading to a more transparent and more easily diag-
nosed reconstruction process. We present PXST results from
three separate experiments, each with a different sample,
effective magnification and defocus distance.
2. Wavefront analysis
The experiment setup and processing pipeline are roughly
equivalent for each experiment, as illustrated in Fig. 2. In this
configuration the sample was placed a distance z1 downstream
of the 2D beam focus, which was formed using two crossed and
wedged MLLs (one MLL to focus vertically and the other
horizontally). The focal length of the lens closest to the focus
was reduced by its distance from the other lens so that the
research papers
928 Andrew J. Morgan et al. � Speckle tracking with multi-layer Laue lens systems J. Appl. Cryst. (2020). 53, 927–936
Figure 1A wedged multi-layer Laue lens of focal length f is constructed fromlayers whose spacing follows the zone-plate condition. To achieve highefficiency the lens must be thick, in which case diffraction is a volumeeffect described by dynamical diffraction. In this case the layers should betilted to locally obey Bragg’s law, which places them normal to a circle ofradius 2f.
2 Methods for dealing with partial coherence, which can be characterized by afew dominant modes, have been successfully developed for ptychography(Thibault & Menzel, 2013; Pelz et al., 2014). Nevertheless, each mode is treatedin a fully coherent way, consistent with the original (single-mode) ptycho-graphic approach.
focal points for the two MLLs meet in the same plane. A total
of N images (In) were then recorded on a detector placed a
distance z downstream of the sample, as the sample was
translated in a 2D grid pattern a distance �xn in the xy plane
(perpendicular to the optical axis for the nth image). If z1 is
sufficiently large, then the images formed on the detector
resemble shadow images of the sample, which are variously
called Gabor or in-line holograms, near-field images, phase
contrast images etc. depending on the specific application and
properties of the sample (for the rest of this article we shall
refer to such images simply as shadow images).
In the ideal case, for a thin sample, a lens system without
any aberrations, ignoring diffraction from a lens aperture and
for large z1, the lens will produce a spherical wavefront and it
can be shown that the observed shadow image will be
equivalent to a defocused and magnified image of the sample
(Iref), such that Inðx; zÞ ¼ M�2Irefðx=M ��xn; zÞ, where the
magnification factor M is given by (z1 + z)/z1 and the effective
defocus z is given by zz1/(z1 + z). Morgan, Quiney et al. (2020)
generalized this principle to incorporate the divergent illu-
mination formed by a non-ideal lens system, so that
InðxÞ ’ WðxÞIref½uðxÞ ��xn; z�; ð1Þ
where W(x) is the ‘white-field image’, the intensity distribution
measured on the detector without the presence of the sample.
u(x) is a 2D vector field that captures both the average
magnification of the image (due to the global phase curvature
of the illumination) and the geometric distortions (arising
from the finite aperture and lens aberrations) in the shadow
image, given by
uðxÞ ¼ x��z
2�r�ðxÞ; ð2Þ
where � is the wavelength of the radiation, r = (@/@x, @/@y) is
the transverse gradient operator and � is the phase of the
wavefield produced by the lens system in the detector plane
(in the absence of the sample).
Using equations (1) and (2) and the set of shadow images
(In), the wavefront formed by the two MLLs in the detector
plane, given by the phase (�) and intensity (W), as well as the
undistorted, magnified and defocused image of the sample,
which we call the ‘reference’ Iref, was recovered by tracking
the local displacement of features formed in each of the
shadow images according to the recipe described by Morgan,
Quiney et al. (2020) using a speckle tracking software package
Figure 2Illustration of the ptychographic XST method. The beamline illuminationwas focused (off-axis) in two dimensions by two crossed and wedgedMLLs. The Siemens star sample was placed 371 mm downstream of thefocal plane. Images were recorded on a pixel array detector 0.71 mdownstream of the sample. The scan data consist of 20 � 20 shadowimages, recorded as the sample was translated across the beam profile.The phase and reference image maps were refined iteratively.
2.1. Image reconstruction with the example of the Siemensstar sample
For this experiment, shadow images of a Siemens star test
sample were recorded at the NSLS-II HXN beamline
(Nazaretski et al., 2014, 2017). Fig. 3 (left) shows one of the 400
shadow images recorded as part of the scan. To achieve a 2D
focus we would ideally use two MLLs, one to focus vertically
and the other horizontally, that are optimized for the same
photon energy. In this experiment, however, we had one lens
that was optimal at 16.7 keV and another at 16.9 keV. We
decided to operate at 16.9 keV. Because of this mismatch of
0.2 keV, the vertically focusing MLL does not focus X-rays
with uniform efficiency across the entire physical aperture.
This results in the tapered fall-off in diffraction intensity near
the top of the figure, corresponding to higher diffraction
angles; the optical axis is located beyond the bottom left of the
figure. The horizontally focusing MLL provided an X-ray
focus with near perfect uniformity across the entire pupil
region along the horizontal direction. In addition to scattering
from the sample and the faint cross-hatch pattern (which we
speculate are due to small local variations in the layer
periods), there are also intensity variations across the image
caused by the non-uniform illumination incident on the MLL
lens system from up-stream optical elements.
The Siemens star test sample has a total diameter of 10 mm
and consists of 30 radial ‘spokes’ with circular cuts at two
radial positions. It is constructed from gold with a projected
thickness in the range 0.5 to 1 mm. The ‘spoke’ tip, facing the
centre of the star, has a width of 100 nm. In order to avoid
speckle registration errors that would arise when translating
features at different camera lengths across the field of view, for
example, features at the top and bottom surface of the sample,
the projected thickness of the sample (�z) should not be
much greater than half the ratio of the demagnified pixel size
(�) to the numerical aperture of the lens system (NA): �z <
�/(2NA). In the present case, � ’ 30 nm and NA ’ 0.017,
leading to the condition �z < 0.9 mm, and so the Siemens
star’s projected thickness is about the tolerable limit of this
method for the current lens system and magnification factor.
Similarly, the plane of translation of the sample must be nearly
parallel to the plane transverse to the optical axis, such that �< �/FOV, where FOV is the field of view, or the side length of
the footprint of the beam on the sample. In the present case,
this limits the tilt angle to less than 24 mrad (to the best of our
knowledge � = 0 during this experiment).
The geometry of the Siemens star helps to visualize the
effect of the low-order aberrations in the lens system on the
observed shadow images. These aberrations led to low-spatial-
frequency geometric distortions that break the approximate
circular symmetry of the image, which is evident in Fig. 1 (left).
To the right we show a magnified view of the region of interest.
Here, we can observe approximately three Fresnel fringes
generated by the sharp outer edges of the Siemens star spokes.
This is the same fringe structure one would observe by illu-
minating the sample with plane wave illumination and
recording an image on a detector placed a distance z ¼ 0:37 m
downstream of the object and magnified by a factor of M =
1917. The effective Fresnel number is then given by
F ¼ X2=ð�zÞ, where X is the full-period spatial frequency of a
feature in the sample. In the present case we have F ¼ 0:18,
corresponding to the smallest width of Siemens star spoke, X =
100 nm.
The white-field image (W) was set to the median value at
each pixel on the detector over the 400 measurements. A more
direct approach would have been to record an image after
completely removing the sample from the incident wavefield.
We found, however, that the former strategy led to superior
results. We speculate that this is due to low-frequency
temporal drifts in either the positioning or the upstream illu-
mination of the MLL system, leading to small variations in the
intensity profile of the beam. Naturally, these drifts also occur
during the acquisition time of the data set and could limit the
viability of this method in cases where the duration of the
experiment far exceeds the duration of stability for the
imaging system.
The initial estimate for the gradient of the wavefield in the
detector plane (r�) was set to
r�ðxÞ ¼2�
�
�x
zx1
;y
zy1
�; ð3Þ
where zx1 and z
y1 are the distances between the sample plane
and the horizontal and vertical focal planes of the lens system,
respectively. Note that for an astigmatic lens system, zx1 6¼ z
y1.
Estimates for zx1 and z
y1 were obtained, in turn, by fitting a set
of parameters in a forward model for the power spectrum of
the data, obtained by summing the mod square of the Fourier
transform of each image. The Fresnel fringes present in each
image produce a nearly circular ring pattern in the cumulative
power spectrum, known as ‘Thon rings’, where the shape and
spacing of the rings provide estimates for defocus and astig-
matism. This algorithm was adapted from the program
CTFFIND4 (Rohou & Grigorieff, 2015), which was developed
for use on cryo-electron microscopy micrographs.
In the top panel of Fig. 4 we show the reconstructed
reference of the sample (Iref). We note that this is not a real-
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930 Andrew J. Morgan et al. � Speckle tracking with multi-layer Laue lens systems J. Appl. Cryst. (2020). 53, 927–936
Figure 3(Left) Raw detector image of the Siemens star shadow (480� 438 pixels).The unfocused beam was blocked by a beam stop placed beyond thebottom left of this figure. The dashed red and black box outlines theregion shown on the right at a higher image magnification. The linearcolour scale is displayed on the far right and ranges from 0 (black) to 3000(white) photon counts.
space image of the sample but rather a magnified view of the
defocused image. Correctly reconstructed, this reference
image will be free of the geometric aberrations present in each
of the measured images, and indeed, this appears to be the
case here. The direct (real-space) imaging resolution is limited
by the effective defocus distance, so that point-like features
will produce overlapping spots at a separation distance less
than 331 nm (Rayleigh criterion), rather than the de-magni-
fied pixel size of 28 nm. This is the separation distance
between the inner edges of the spokes of the Siemens star
when the first minimum of the edge’s Fresnel fringes overlaps
with the brighter zeroth-order maximum of the adjacent edge.
Another measure of resolution is the Fourier power spectrum
(FPS) cut-off frequency, which is given by the highest spatial
frequencies in an image above the signal-to-noise level. The
FPS is graphed in Fig. 5 (top panel), with the vertical black line
indicating the full-period resolution of the image at 70 nm, or
a half-period resolution of 35 nm, approximately 20% greater
than the de-magnified pixel size.
In Fig. 6 we show two real-space reconstructions of the
Siemens star’s projected mass (bottom left and right panels).
For a sample constructed from a single material, with a
constant density, and a linear approximation to Beer’s law, the
projected mass is proportional to the thickness, or the height
of the sample above the substrate. Both were recovered from
the reference (top-left panel) and can be compared with a raw
diffraction image, shown in the top-right panel. In the bottom
row, we display the thickness profile recovered via the trans-
port of intensity equation (TIE) and via contrast transfer
function (CTF) inversion, in the left and right panels,
respectively, using the X-TRACT software package (Gureyev
et al., 2011). We note that neither method is ideal in the
present case: the TIE algorithm works best for large Fresnel
numbers and the CTF inversion is ideal for weak phase
objects. Nevertheless, the ends of the ‘spokes’ near the centre
of the Siemens star, with a separation distance �158 nm, can
clearly be distinguished in both images, which is an improve-
ment on the direct (real-space) resolution of the reference.
The vector field u(x) � �xn defines the mapping between
each point in the nth image [In(x)] and a point in the reference
[see equation (1)]. The phase gradients were obtained from u
via equation (2), using the formalism described by Morgan,
Quiney et al. (2020), and are shown in the bottom-left panel of
Fig. 4. Here we display the phase gradients, after removing the
global shift and magnification factors, as a black quiver plot,
scaled to pixel units. In order to further illustrate the effect of
research papers
J. Appl. Cryst. (2020). 53, 927–936 Andrew J. Morgan et al. � Speckle tracking with multi-layer Laue lens systems 931
Figure 4(Top) Reference image (Iref) of the Siemens star test sample. (Bottomleft) Phase profile of the pupil function � (colour scale), overlayed with aquiver plot of the retrieved phase gradient r� vector field (scaled topixel units). (Bottom right) Four views of the central region of theSiemens star. In the top right is the undistorted view (as outlined in blackin the top panel). The remaining three panels show this feature as itappears in different locations on the detector array [after division byW(x)] corresponding to the regions indicated by like-coloured outlines inthe left panel.
Figure 5(Top) The azimuthal average of the Fourier power spectrum of therecovered reference image of the Siemens star sample. The FPS isobtained by taking the mod square of the Fourier transform of Iref. Theblue dashed line shows the noise floor, which was estimated by taking theaverage of the FPS over the last 30 values. The resolution cut-off (greyvertical line) is given by the resolution at which the FPS is equal to twicethe noise floor (black dashed line). (Bottom) Histogram of the differencebetween the recovered wavefront angles (detector plane) from each ofthe split-1/2 data sets (blue bar chart). The solid black line shows theGaussian model fit with a standard deviation of 6.0 nrad.
the geometric aberrations, beyond the overall magnification,
caused by the phase gradients in the lens phase profile, we
display a magnified view of the central region of the Siemens
star (see the black box in the top panel of Fig. 4) as it appears
in three different shadow images. The regions within the white
field where this feature appears for each of the three images
are illustrated by the coloured square outlines shown in the
bottom-left panel. In the bottom-right panel we show the
corresponding regions for each of these images. To increase
the contrast, we have divided the images by the white-field
image (In/W). In the top-right sub-panel, we also show the
same region of the recovered reference. Here one can clearly
observe local variations in the degree of magnification, along
the x and/or y axis, depending on the position of the sample
within the incident wavefield.
In the bottom-left panel of Fig. 4 we show the residual phase
profile of the MLL lens system (colour map). The residual
phase profile is obtained after removing the constant, linear
and quadratic components of the global phase profile, which
correspond to an overall phase constant, a tilt term and the
defocus aberrations, respectively. By removing these terms, it
is possible to perceive the small deviations in the phase from
an (ideal) quadratic profile. Armed with this phase profile, we
could then numerically propagate the wavefield to the region
near the focal plane of the lens, as shown in Fig. 7. These
results were obtained after three iterations of the PXST
update algorithm. For each iteration, we refined the initial
estimates for the sample stage translations. The ‘irrotational
constraint’ on the phase gradients was also enforced [see
Section 5 of Morgan, Quiney et al. (2020)].
In Appendix A we show a comparison of the recovered
wavefront phase from a separate PXST experiment and a
ptychographic experiment taken with the sample placed
nearer to the focal plane. Both results show qualitative
agreement; however, the root-mean-squared difference is
much greater than we would predict if the ptychographic
result is considered to be the ground truth.
The local angular distribution of the wavefront rays, in the
plane of the detector, is given by � ¼ ð�=2�Þr�. In the ideal
case, the smallest resolvable angular deviation of a ray (the
angular sensitivity) is given by �� = �pix�det/zM, where �det is
the width of the point spread function of the detector (greater
than or equal to the physical pixel size) and �pix is the frac-
tional reduction in the effective pixel size due to numerical
interpolation. In the present case, setting �det’ 55 m and �pix <
1, we have �� < 40 nrad.
In order to estimate the achieved angular resolution, we
randomly assigned each pixel of each image to one of two data
sets. Keeping the reconstructed reference map and sample
stage positions from the original reconstruction, we then
repeated the reconstruction of the phase gradients indepen-
dently for each of the two data sets. This process is only
possible because of the high degree of redundancy in the
original data. A histogram of the difference between the two
reconstructions, shown in the bottom panel of Fig. 5, provides
an estimate for the underlying uncertainty in the recovered �values. The standard deviation of the difference �1 � �2
yields ��’ 6.0 nrad, which suggests a �pix value of less than 2/
10. This shows that from the redundancy of data, caused by
measurements at any given location in the wave with many
positions of the object, one is able to interpolate angular
deviations to a small fraction of a pixel. The angular distri-
bution is related to the phase profile via 2D integration,
research papers
932 Andrew J. Morgan et al. � Speckle tracking with multi-layer Laue lens systems J. Appl. Cryst. (2020). 53, 927–936
Figure 7Projection of the wavefront intensity profile near the focal plane of thelens system, along the x axis (top) and the y axis (bottom). The linearcolour scale ranges from 0 (white) to 1 (black) in arbitrary units.
Figure 6(Top left) Subregion of the reference reconstruction. The linear colourscale ranges from 0 (black) to 1.6 (white). (Top right) Subregion of image250 in the data set, without any preprocessing. The linear colour scaleranges from 0 (black) to 4000 (white) photon counts. (Bottom left) TIEreconstruction of the same subregion as in the top-left panel. (Bottomright) CTF inverted reconstruction of the same region. The colour scale isthe same as in the bottom-left panel.
�ðxÞ ¼ ð2�=�ÞR R
�ðxÞ dx, and propagating the uncertainties
yields an estimate for the phase sensitivity of �� ’ 0.065 rad
(0.01 waves).
2.2. Diatom sample
For this experiment, the biomineralized shell of a marine
planktonic diatom was placed on a silicon nitride membrane
and scanned across the wavefield 2.22 mm downstream of the
lens focus. In contrast to the Siemens star experiment, the
effective defocus and magnification (see Table 1) are such that
only first-order Fresnel fringes are visible across the majority
of the reference. For this reason we did not use the Thon rings
to provide initial estimates for zx1 and z
y1. Instead, we set
z1 ¼ zx1 ¼ z
y1 in equation (3) and chose the value of z1 which
minimized the sum squared error after many trials over a
range of z1 values. Errors in the initial estimates for z1, zx1 and
zy1 will lead to additional defocus aberrations in the recovered
phase map, which can then be removed as needed. If these
errors are too large, however, the algorithm may take many
more iterations (or fail completely) to converge.
In contrast to the previous experiment, only a fraction
(roughly 1/9th) of the object is visible in the field of view for
each image. The reference image is shown in Fig. 8, obtained
after three iterations of the PXST algorithm.
This diatom was collected from the Antarctic sea and its
shell is made from a complex network of nanostructured silica
with an exceptional strength-to-weight ratio, despite being
produced under low temperature and pressure conditions. The
circular shell of the diatom is constructed from six azimuthal
segments, which extend in a dome-like fashion out of the page
for the orientation shown in Fig. 8. The boundary of these
segments can be observed as six radial creases, extending from
the edge of the inner circle to the outer rim of the sample. This
sixfold symmetry is a motif that is repeated throughout the
diatom structure: see for example the approximate hexagonal
packing of the small ‘white dots’ with a diameter of about
5 mm. In another scan (discussed in the next section), taken
with the sample closer to the focus, a more detailed view of
these ‘white dots’ can be seen. This more magnified view of the
diatom is displayed in the top-right corner of the figure, and
one can see that these dots are themselves hexagonal in shape
with what appear to be hollow depressions in the centre.
The estimated angular sensitivity for this reconstruction is
20 nrad, which is approximately 3.2 times greater than for the
Siemens star reconstruction. This result is consistent with the
corresponding decrease in the average magnification by a
factor of 3.3, from 1917 (Siemens star) to 595 (diatom). The
direct (real-space) imaging resolution was 410 nm (Rayleigh
criterion), while the FPS cut-off frequency was 259 nm, with a
half-period resolution of 130 nm, which is 40% greater than
the de-magnified pixel size.
2.3. Diatom subregion
For this experiment the sample was moved closer to the
focal plane of the lens, from 2.22 mm in the previous section to
0.57 mm here. This corresponds to an increase in the magni-
fication by a factor of 3.9, from 595 to 2308. As discussed by
Morgan, Quiney et al. (2020), the upper limit to the magnifi-
cation factor for this particular technique is governed by the
smallest distance between the focal plane and the sample such
that the diffraction remains in the near-field imaging regime.
For larger magnification factors, with the sample closer to the
focal plane, the rapidly oscillating phase and intensity of the
illuminating wavefield lead to significant errors in the speckle
tracking approximation of equation (1). Here, however,
another difficulty was encountered, relating to the pseudo
translational symmetry of the diatom structure at this
magnification.
The FPS of the reference, in the top panel of Fig. 9, shows
an hexagonal array of points overlaid on top of the much
weaker Thon rings, which (again) arise because the reference
is a defocused image of the sample’s exit-surface wave. The
locations of the peaks reveal the reciprocal lattice of the real-
space structure, which is approximately hexagonal with a
primitive lattice constant of �601 nm. This approximate
translational symmetry is undesirable in PXST because of the
possibility of miss-registering features between each recorded
image and the reference by an amount equal to the lattice
constant.
In the bottom-left panel of Fig. 10 we show a failed
reconstruction of the pixel mapping between the recorded
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J. Appl. Cryst. (2020). 53, 927–936 Andrew J. Morgan et al. � Speckle tracking with multi-layer Laue lens systems 933
Figure 8The diatom’s reference image. The linear greyscale colour map rangesfrom 0.92 (black) to 1.08 (white). The reconstructed area outside of thediatom’s region of interest has been masked. The demagnified pixel areais 93 � 93 nm. The field of view of the image is 122 � 120 mm(1320 � 1290 pixels). Fine details in the sub-structure of the diatom arevisible in this phase-contrast projection image, which are otherwiseobscured by the surface of the sample in scanning electron micrographimages. (Top right) Magnified image map of a subregion of the diatom.The field of view is 95 � 107 mm (414 � 466 pixels), with a demagnifiedpixel area of 24 � 24 nm. The small blue rectangle indicates the scale ofthe inset with respect to the larger image of the diatom.
images (one of which is shown in the top-left panel) and the
reference. At the bottom of the image one can see a horizontal
step-like reduction in the mapping function from white to
black, corresponding to a reduction of 20 pixels. When scaled
to physical units, this drop corresponds exactly to the hexa-
gonal lattice spacing of the diatom substructure. In order to
overcome this problem, we chose to regularize the recovered
pixel shifts by convolving them with a Gaussian kernel at each
iteration. The standard deviation of this kernel was reduced
linearly from 20 pixels to 0 pixels as the iterations progressed.
In this way sharp deviations in the mapping function were
prevented from forming early in the reconstruction process.
The result of this regularization procedure is shown in the
bottom-right panel of the figure, where the step-like artefact is
no longer present in the reconstructed pixel mapping.
3. Discussion and conclusion
In this article we have demonstrated the use of PXST on three
experimental data sets. In each case, both the illuminating
wavefront and a highly magnified, undistorted in-line holo-
gram of the sample were recovered. The main benefit of PXST
over other speckle tracking techniques, for example, the
unified modulated pattern analysis (UMPA) approach of
Zdora et al. (2017), the geometric flow algorithm of Paganin et
al. (2018) and the original XST technique of Berujon et al.
(2012), is that it is able to deal with highly divergent illumi-
nation. This allows for comparatively large magnification
factors (e.g. 2308 for the diatom subregion), which leads to a
corresponding increase in the achievable ray angle sensitivity
(3.4 nrad) and image resolution (45 nm full period). Conver-
sely, PXST does not provide a direct (real-space) image of the
sample’s phase, absorption or ‘dark-field’ profiles.
Another approach that is suitable for highly divergent
illumination is the X-ray speckle scanning technique of
Berujon et al. (2012), which provides a phase sensitivity
proportional to the step size of the sample translations. In
PXST, however, the phase sensitivity does not depend on the
step size, making it suitable for a broader range of experiment
facilities.
With the high-NA, efficient, hard X-ray optics provided by
the wedged MLLs used here, the footprint of the beam on the
sample is greater for a fixed magnification factor than would
otherwise be the case. This increases the throughput of the
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934 Andrew J. Morgan et al. � Speckle tracking with multi-layer Laue lens systems J. Appl. Cryst. (2020). 53, 927–936
Figure 9(Top) Image of the FPS of the diatom subregion. The full-period spatialfrequency at the left edge of the image is 48 nm�1. To avoid artefacts fromthe sharp edges of the real-space image (as shown in the subpanel ofFig. 8) the FPS was filtered with a Gaussian window function with astandard deviation of 2.4 mm. Before display, the FPS was raised to thepower 0.1, in order to reveal the Thon rings underneath the muchstronger peaks from the hexagonal lattice. (Bottom) Azimuthal averageof the FPS, with a cut-off frequency corresponding to a full-periodresolution of 45 nm (half-period resolution of 22.5 nm, 5% smaller thanthe de-magnified pixel size).
Figure 10(Top left) Image 50 of the 121 recorded shadow images. This image spansdiffraction angles of 15 � 17 mrad. The linear colour scale ranges from 0(black) to 2000 (white) photon counts. (Top right) The same imagedivided by the white field (W); the colour scale ranges from 0.9 to 1.2.(Bottom left) The recovered pixel mapping between the recorded imagesand the reference image u(x) (in pixel units); the colour scale ranges from�10 (black) to 10 (white) pixel shifts. (Bottom right) The recovered pixelmapping when employing regularization during the reconstruction; thesame colour scale as bottom left.
imaging method, by a factor proportional to the square of the
increase in the NA.
We have also demonstrated that PXST does not require an
additional diffuser in the beam path and we expect that a wide
variety of samples could be used as a wavefront-sensing device
– although a dense random object such as a diffuser should
reduce the number of required images.
In future, we hope to develop the PXST algorithm for use in
‘cone-beam tomography’, a geometry where the illumination
diverges significantly as it passes through the object.
APPENDIX AComparison of the recovered wavefront viaptychography and PXST
In Fig. 11 we show the phase of the wavefront recovered via
far-field ptychography (left) and PXST (right), from two
independent data sets obtained at the European Synchrotron
Radiation Facility.31In both experiments, a Siemens star test
sample was scanned in a 2D grid pattern across the wavefront,
with a focus-to-sample distance of 0.13 m. For the ptycho-
graphic data set, the sample was scanned near the focal plane
of the lens (z1 = 1.01 mm) and the angular extent of the
diffraction extended well beyond that of the diverging illu-
mination, i.e. outside of the holographic region. For the PXST
data set, the sample was placed further from the focus (with
z1 = 5.8 mm) and the diffraction was predominantly confined
to the holographic region of the detector (in a 2 � 1.6 mrad
angular window), consistent with the near-field scattering
regime.
One advantage of PXST over ptychography is that the
phase profile is not ‘wrapped’ onto the [��, �) domain. This is
useful in cases where the intent is for the recovered phases to
inform a structural analysis of the lens system, such as the
height of a mirror surface or the local period of bilayers in an
MLL. In some cases, however, the phases recovered from
ptychography can be ‘unwrapped’; for smooth phase profiles,
continuity of the phases allows one to identify regions
bounded by discontinuous 2� phase jumps. One can then add
or subtract 2� to the phases in these regions as needed until
the entire phase profile is smooth. This procedure was applied
to unwrap the phases shown in the left panel.
The two phase profiles show qualitative agreement between
the ptychographic and PXST algorithms. The root-mean-
squared deviation is �5 rad, which is many orders of magni-
tude worse than the theoretically achievable phase sensitivity.
Therefore, one or both of the reconstructions suffers from
systematic artefacts in the recovered phases. This is a matter
for further investigation.
Acknowledgements
We acknowledge Lars Gumprecht, Julia Maracke, Siegfried
Imlau (CFEL), Sabrina Bolmer, Sven Korseck, Janning
Meinert, Florian Pithan, David Pennicard, Andre Rothkirch
and Heinz Graafsma (DESY) for various contributions.
Andrzej Andrejczuk (University of Bialystok, Poland)
assisted with theoretical aspects of the propagation of light
through the MLLs. Timur E. Gureyev assisted with the TIE-
and CTF-based reconstructions of the Siemens star. Christian
Hamm, from the Alfred Wegener Institute, Helmholtz Centre
for Polar and Marine Research, provided the diatom sample.
KS acknowledges the Joachim-Herz Stiftung.
Funding information
Funding for this project was provided by the Australian
Research Council Centre of Excellence in Advanced Mol-
ecular Imaging (AMI), the Gottfried Wilhelm Leibniz
Program of the Deutsche Forschungsgemeinschaft (DFG), the
NSF award 1231306 and the Cluster of Excellence ‘CUI:
Advanced Imaging of Matter’ of the DFG – EXC 2056 –
project ID 390715994. This research used the HXN beamline
of the National Synchrotron Light Source II, a US Depart-
ment of Energy (DOE) Office of Science User Facility
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J. Appl. Cryst. (2020). 53, 927–936 Andrew J. Morgan et al. � Speckle tracking with multi-layer Laue lens systems 935
Figure 11Phase of the recovered wavefronts via ptychography (left) and PXST (right). The ptychographic phase profile was unwrapped before display. The colourscale is in radian units.
3 Further analysis using these data sets is provided by Murray et al. (2019).
operated for the DOE Office of Science by Brookhaven
National Laboratory under contract No. DE-SC0012704.
References
Bajt, S., Prasciolu, M., Fleckenstein, H., Domaracky, M., Chapman,H. N., Morgan, A. J., Yefanov, O., Messerschmidt, M., Du, Y.,Murray, K. T., Mariani, V., Kuhn, M., Aplin, S., Pande, K.,Villanueva-Perez, P., Stachnik, K., Chen, J. P. J., Andrejczuk, A.,Meents, A., Burkhardt, A., Pennicard, D., Huang, X., Yan, H.,Nazaretski, E., Chu, Y. S. & Hamm, C. E. (2018). Light Sci. Appl. 7,17162.
Berujon, S., Wang, H. & Sawhney, K. (2012). Phys. Rev. A, 86, 063813.Chapman, H. N. (1996). Ultramicroscopy, 66, 153.Gureyev, T. E., Nesterets, Y., Ternovski, D., Thompson, D., Wilkins,
S. W., Stevenson, A. W., Sakellariou, A. & Taylor, J. A. (2011). Proc.SPIE, 8141, 81410B.
Maia, F. R. N. C. (2012). Nat. Methods, 9, 854–855.Morgan, A. J., Murray, K. T., Quiney, H. M., Bajt, S. & Chapman, H. N.
(2020). J. Appl. Cryst. 53. Submitted.Morgan, A. J., Prasciolu, M., Andrejczuk, A., Krzywinski, J., Meents,
A., Pennicard, D., Graafsma, H., Barty, A., Bean, R. J.,Barthelmess, M., Oberthuer, D., Yefanov, O. M., Aquila, A.,Chapman, H. N. & Bajt, S. (2015). Sci. Rep. 5, 9892.
Morgan, A. J., Quiney, H. M., Bajt, S. & Chapman, H. N. (2020). J.Appl. Cryst. 53, 760–780.
Morgan, K. S., Paganin, D. M. & Siu, K. K. W. (2012). Appl. Phys.Lett. 100, 124102.
Murray, K. T., Pedersen, A. F., Mohacsi, I., Detlefs, C., Morgan, A. J.,Prasciolu, M., Yildirim, C., Simons, H., Jakobsen, A. C., Chapman,H. N., Poulsen, H. F. & Bajt, S. (2019). Opt. Express, 27, 7120.
Nazaretski, E., Huang, X., Yan, H., Lauer, K., Conley, R., Bouet, N.,Zhou, J., Xu, W., Eom, D., Legnini, D., Harder, R., Lin, C. H., Chen,Y. S., Hwu, Y. & Chu, Y. S. (2014). Rev. Sci. Instrum. 85, 033707.
Nazaretski, E., Yan, H., Lauer, K., Bouet, N., Huang, X., Xu, W.,Zhou, J., Shu, D., Hwu, Y. & Chu, Y. S. (2017). J. Synchrotron Rad.24, 1113–1119.
Paganin, D. M., Labriet, H., Brun, E. & Berujon, S. (2018). Phys. Rev.A, 98, 053813.
Pelz, P. M., Guizar-Sicairos, M., Thibault, P., Johnson, I., Holler, M. &Menzel, A. (2014). Appl. Phys. Lett. 105, 251101.
Prasciolu, M., Leontowich, A. F. G., Krzywinski, J., Andrejczuk, A.,Chapman, H. N. & Bajt, S. (2015). Opt. Mater. Expr. 5, 748.
Rodenburg, J. M., Hurst, A. C. & Cullis, A. G. (2007). Ultramicro-scopy, 107, 227–231.
Rohou, A. & Grigorieff, N. (2015). J. Struct. Biol. 192, 216–221.Thibault, P., Dierolf, M., Bunk, O., Menzel, A. & Pfeiffer, F. (2009).
Ultramicroscopy, 109, 338–343.Thibault, P. & Menzel, A. (2013). Nature, 494, 68–71.Yan, H., Conley, R., Bouet, N. & Chu, Y. S. (2014). J. Phys. D Appl.
Phys. 47, 263001.Zdora, M.-C. (2018). J. Imaging, 4, 60.Zdora, M. C., Thibault, P., Zhou, T., Koch, F. J., Romell, J., Sala, S.,
Last, A., Rau, C. & Zanette, I. (2017). Phys. Rev. Lett. 118, 203903.
research papers
936 Andrew J. Morgan et al. � Speckle tracking with multi-layer Laue lens systems J. Appl. Cryst. (2020). 53, 927–936
